1. About this Course Subject: Textbook Reference book Course website
Digital Signal Processing
EE 541
Textbook
Discrete Time Signal Processing
A. V. Oppenheim and R. W. Schafer, Prentice Hall, 3rd Edition
Reference book
Probability and Random Processes with Applications to Signal Processing
Henry Stark and John W. Woods, Prentice Hall, 3rd Edition
Course website
Syllabus, lecture notes, homework, solutions etc.

2. About this Course Grading details: Homework: (Weekly) 20% Midterm: 30%
Final: 30%
Project: 20%

3. Matlab
Powerful software you will like for the rest of your time in ShanghaiTech SIST
Ideal for practicing the concepts learnt in this class and doing the final projects

4. About the Lecturer Name: Xiliang Luo (罗喜良) Research interests:
Wireless communication
Signal processing
Information theory
More information:

5. About TA
Name: 裴东 Contact: Office hour: Friday, 6-8pm,

6. Some survey background coolest thing you have ever done
what you want to learn from this course?

7. Lecture 1: Introduction to DSP
Xiliang Luo
2014/9

8. Signals and Systems Signal something conveying information
speech signal
video signal
communication signal
continuous time
discrete time
digital signal : not only time is discrete, but also is the amplitude!

9. Discrete Time Signals
Mathematically, discrete-time signals can be expressed as a sequence of numbers
𝑥 𝑛 ,𝑛∈𝑍
In practice, we obtain a discrete-time signal by sampling a continuous- time signal as:
𝑥 𝑛 = 𝑥 𝑎 (𝑛𝑇)
where T is the sampling period and the sampling frequency is defined as 1/T

10. Speech Signal Question: 1. What is the sampling frequency?
2. Are we losing anything here by sampling?

11. Some Basic Sequences Unit Sample Sequence 𝛿 𝑛 = 0, 𝑛≠0 1, 𝑛=0
𝛿 𝑛 = 0, 𝑛≠0 1, 𝑛=0
Unit Step Sequence
𝛿 𝑛 = 0, 𝑛<0 1, 𝑛≥0

12. Some Basic Sequences Sinusoidal Sequence x 𝑛 =𝐴 cos ( 𝜔 0 𝑛+𝜙)
Question:
Is discrete sinusoidal periodic?
What is the period?
Question:
Cos(pi/4xn) vs Cos(7pi/4xn), which
One has faster oscillation?

13. Some Basic Sequences Sinusoidal Sequence x 𝑛 =𝐴 cos ( 𝜔 0 𝑛+𝜙)
Question:
Cos(pi/4xn) vs Cos(7pi/4xn), which
One has faster oscillation?

14. Discrete-Time Systems
a transformation or operator mapping discrete time input to discrete time output
𝑦 𝑛 =𝑇{𝑥[𝑛]}
Example: ideal delay system
y[n] = x[n-d]
Example: moving average
y[n] = average{x[n-p],….,x[n+q]}

15. Memoryless System
Definition: output at time n depends only on the input at the sample time n
𝑦 𝑛 =𝑥 𝑛 2
Question:
Are the following memoryless?
y[n] = x[n-d]
y[n] = average{x[n-p], …, x[n+q]}

16. Linear System
Definition: systems satisfying the principle of superposition
𝑇 𝑥 1 𝑛 + 𝑥 2 𝑛 =𝑇 𝑥 1 [𝑛] +𝑇{ 𝑥 2 [𝑛]}
𝑇 𝑎𝑥[𝑛] =𝑎𝑇 𝑥[𝑛]
𝑇 𝑎 𝑥 1 𝑛 + 𝑏𝑥 2 𝑛 =𝑎𝑇 𝑥 1 [𝑛] +𝑏𝑇{ 𝑥 2 [𝑛]}
Additivity Property
Scaling Property
Superposition Principle

17. Time-Invariant System
A.k.a. shift-invariant system: a time shift in the input causes a corresponding time shift in the output:
𝑇 𝑥[𝑛] =𝑦[𝑛]
𝑇 𝑥[𝑛−𝑑] =𝑦[𝑛−𝑑]
Question:
Are the following time-invariant?
y[n] = x[n-d]
y[n] = x[Mn]

18. Causality
The output of the system at time n depends only on the input sequence at time values before or at time n;
Is the following system causal?
y[n] = x[n+1] – x[n]

19. Stability: BIBO Stable
A system is stable in the Bounded-Input, Bounded-Output (BIBO) sense if and only if every bounded input sequence produces a bounded output sequence.
A sequence is bounded if there exists a fixed positive finite value B such that:
𝑥 𝑛 ≤𝐵<∞

20. LTI Systems LTI : both Linear and Time-Invariant systems
convenient representation: completely characterized by its impulse response
significant signal-processing applications
Impulse response
LTI System
ℎ 𝑛 =𝑇{𝛿[𝑛]}
𝑥 𝑛 = 𝑘 𝑥 𝑘 𝛿[𝑛−𝑘]
𝑦 𝑛 =𝑇 𝑘 𝑥 𝑘 𝛿[𝑛−𝑘] = 𝑘 𝑥 𝑘 𝑇{𝛿 𝑛−𝑘 ] = 𝑘 𝑥 𝑘 ℎ[𝑛−𝑘]

21. LTI System
LTI system is completely characterized by its impulse response as follows:
ℎ 𝑛 =𝑇{𝛿[𝑛]}
𝑦 𝑛 = 𝑘 𝑥 𝑘 ℎ 𝑛−𝑘
convolution sum
≜𝑥 𝑛 ∗ℎ[𝑛]

22. Properties of LTI Systems
Commutative:
Distributive:
Associative:
𝑥 𝑛 ∗ℎ 𝑛 =ℎ 𝑛 ∗𝑥[𝑛]
𝑥 𝑛 ∗ ℎ 1 𝑛 + ℎ 2 𝑛 =𝑥 𝑛 ∗ ℎ 1 𝑛 +𝑥 𝑛 ∗ ℎ 2 [𝑛]
(𝑥 𝑛 ∗ ℎ 1 𝑛 )∗ ℎ 2 𝑛 =𝑥 𝑛 ∗( ℎ 1 𝑛 ∗ ℎ 2 [𝑛])

23. Properties of LTI Systems
Equivalent systems:

24. Properties of LTI Systems
Equivalent systems:

25. Stability of LTI System
LTI systems are stable if and only if the impulse response is absolutely summable:
sufficient condition
need to verify bounded input will have also bounded output under this condition
necessary condition
need to verify: stable system  the impulse response is absolutely summable
equivalently: if the impulse response is not absolutely summable, we can prove the system is not stable!
𝑘=−∞ +∞ |ℎ[𝑘]| <∞

26. Stability of LTI System
Prove: if the impulse response is not absolutely summable, we can define the following sequence:
x[n] is bounded clearly
when x[n] is the input to the system, the output can be found to be the following and not bounded:
𝑥 𝑛 = ℎ ∗ [−𝑛] |ℎ[−𝑛]| , ℎ −𝑛 ≠0 0, ℎ −𝑛 =0
𝑦 0 = 𝑥 𝑘 ℎ −𝑘 = ℎ 𝑘 2 |ℎ[𝑘]|

27. Some Convolution Examples
Matlab cmd: conv()
what is the resulting shape?

28. Some Convolution Examples
what is the resulting shape?

29. Some Convolution Examples
what is the freq here?
sin⁡( 𝑛𝜋 8 )

30. Frequency Domain Representation
Eigenfunction for LTI Systems
complex exponential functions are the eigenfunction of all LTI systems
𝑦 𝑛 = 𝑒 𝑗𝜔𝑛 ∗ℎ 𝑛 = 𝑘 ℎ 𝑘 𝑒 𝑗𝜔 𝑛−𝑘 = 𝑒 𝑗𝜔𝑛 × 𝑘 ℎ 𝑘 𝑒 −𝑗𝜔𝑘
𝐻( 𝑒 𝑗𝜔 )= 𝑘 ℎ 𝑘 𝑒 −𝑗𝜔𝑘
𝑦 𝑛 =𝐻 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛

31. Frequency Response of LTE Systems
For an LTI system with impulse response h[n], the frequency response is defined as:
In terms of magnitude and phase:
𝐻( 𝑒 𝑗𝜔 )= 𝑘 ℎ 𝑘 𝑒 −𝑗𝜔𝑘
phase response
𝐻( 𝑒 𝑗𝜔 )= 𝐻 𝑒 𝑗𝜔 𝑒 ∠𝐻( 𝑒 𝑗𝜔 )
magnitude response

32. Frequency Response of Ideal Delay
ℎ 𝑛 =𝛿[𝑛− 𝑛 𝑑 ]
𝐻 𝑒 𝑗𝜔 = 𝑛 𝛿 𝑛− 𝑛 𝑑 𝑒 −𝑗𝜔𝑛 = 𝑒 −𝑗𝜔 𝑛 𝑑

33. Frequency Response for a Real IR
For real impulse response, we can have:
Response to a sinusoidal of an LTI with real impulse response
𝐻 𝑒 −𝑗𝜔 = 𝐻 ∗ ( 𝑒 𝑗𝜔 )
why?
𝑥 𝑛 = Acos ( 𝜔 0 𝑛+𝜙 ) = 𝐴 2 𝑒 𝑗(𝜙+ 𝜔 0 𝑛) + 𝐴 2 𝑒 −𝑗(𝜙+ 𝜔 0 𝑛)
𝑦 𝑛 = 𝐴 2 𝐻 𝑒 𝑗 𝜔 𝑜 𝑒 𝑗(𝜙+ 𝜔 0 𝑛) + 𝐴 2 𝐻( 𝑒 −𝑗 𝜔 0 ) 𝑒 −𝑗(𝜙+ 𝜔 0 𝑛)
= 𝐴 2 |𝐻 𝑒 𝑗 𝜔 𝑜 | 𝑒 𝑗 𝜙+ 𝜔 0 𝑛+∠𝐻 𝑒 𝑗 𝜔 𝑜 𝐴 2 |𝐻 𝑒 𝑗 𝜔 0 | 𝑒 −𝑗(𝜙+ 𝜔 0 𝑛+∠𝐻 𝑒 𝑗 𝜔 𝑜 )
= 𝐻 𝑒 𝑗 𝜔 𝑜 𝐴 cos ( 𝜔 0 𝑛+𝜙+∠𝐻( 𝑒 𝑗 𝜔 0 ))

34. Frequency Response Property
Frequency response is periodic with period 2π
fundamentally, the following two discrete frequencies are indistinguishable
𝜔, 𝜔+2𝜋
We only need to specify frequency response over an interval of length 2π : [- π, + π];
In discrete time:
low frequency means: around 0
high frequency means: around +/- π

35. Frequency Response of Typical Filters
low pass
band-stop
high pass
band-pass

36. Representation of Sequences by FT
Many sequences can be represented by a Fourier integral as follows:
x[n] can be represented as a superposition of infinitesimally small complex exponentials
Fourier transform is to determine how much of each frequency component is used to synthesize the sequence
𝑥 𝑛 = 1 2𝜋 −𝜋 𝜋 𝑋 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛 𝑑𝜔
Synthesis: Inverse Fourier Transform
Prove it!
𝑋 𝑒 𝑗𝜔 = 𝑛 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛
Analysis: Discrete-Time Fourier Transform

37. Convergence of Fourier Transform
A sufficient condition: absolutely summable
it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function

38. Square Summable A sequence is square summable if:
For square summable sequence, we have mean-square convergence:
𝑛=−∞ ∞ 𝑥[𝑛] 2 <∞

39. Ideal Lowpass Filter

40. DTFT of Complex Exponential Sequence
Let a Fourier Transform function be:
Now, let’s find the synthesized sequence with the above Fourier Transform:

41. Symmetry Properties of DTFT
Conjugate Symmetric Sequence
Conjugate Anti-Symmetric Sequence
Any sequence can be expressed as the sum of a CSS and a CASS as
𝑥 𝑒 𝑛 = 𝑥 𝑒 ∗ [−𝑛]
Real  even sequence
𝑥 𝑜 𝑛 = −𝑥 𝑜 ∗ [−𝑛]
Real  odd sequence
𝑥 𝑛 =𝑥 𝑒 𝑛 + 𝑥 𝑜 [𝑛]
How?

42. Symmetry Properties of DTFT
DTFT of a conjugate symmetric sequence is conjugate symmetric
DTFT of a conjugate anti-symmetric sequence is conjugate anti- symmetric
Any real sequence’s DTFT is conjugate symmetric

43. Fourier Transform Theorems
Time shifting and frequency shifting theorem
Prove it!

44. Fourier Transform Theorems
Time Reversal Theorem
Prove it!

45. Fourier Transform Theorems
Differentiation in Frequency Theorem
Prove it!

46. Fourier Transform Theorems
Parseval’s Theorem: time-domain energy = freq-domain energy
HW Problem 2.84: Prove a more general format

47. Fourier Transform Theorems
Convolution Theorem
Prove it!

48. Fourier Transform Theorems
Windowing Theorem
Prove it!

49. Discrete-Time Random Signals
Wide-sense stationary random process (assuming real)
Consider an LTE system, let x[n] be the input, which is WSS, the output is denoted as y[n], we can show y[n] is WSS also
𝜙 𝑥𝑥 𝑛,𝑚 =𝐸 𝑥 𝑛 𝑥[𝑛+𝑚] = 𝜙 𝑥𝑥 [𝑚]
autocorrelation function

50. Discrete-Time Random Signals
WSS in, WSS out

51. Discrete-Time Random Signals
WSS in, WSS out

52. Discrete-Time Random Signals
WSS in, WSS out

53. Power Spectrum Density
band-pass

54. White Noise
Very widely utilized concept in communication and signal processing
A white noise is a signal for which:
From its PSD, we can see the white noise has equal power distribution over all frequency components
Often we will encounter the term: AWGN, which stands for: additive white Gaussian noise
the underlying random noise is Gaussian distributed
𝜙 𝑥𝑥 𝑚 = 𝜎 𝑥 2 𝛿[𝑚]
Φ 𝑥𝑥 𝑒 𝑗𝜔 = 𝜎 𝑥 2

55. Review LTI system Frequency Response Impulse Response Causality
Stability
Discrete-Time Fourier Transform
WSS
PSD

56. Homework Problems
2.11 Given LTI frequency response, find the output when input a sinusoidal sequence …
2.17 Find DTFT of a windowed sequence …
2.22 Period of output given periodic input …
2.40 Determine the periodicity of signals …
2.45 Cascade of LTE systems …
2.51 Check whether system is linear, time-invariant …
2.63 Find alternative system …
2.84 General format of Parseval’s theorem …
Try to use Matlab to plot the sequences and results when required

57. Next Week
Z – Transform
Please read the textbook Chapter 3 in advance!